Loading packages

Installing a package downloads it to your computer, but doesn’t make it immediately available to your R session. If you want to use the tidyverse package, you will need to load it by running the following code:

library(tidyverse)

(The code is similar for other packages, just replace the name tidyverse). You will need to include library(...) calls in the setup chunk of your R Markdown documents.

Data size and variables

To peak at the data, we can use the glimpse function to see the variables and the size:

glimpse(penguins)

## Rows: 344
## Columns: 8
## $ species           <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adel…
## $ island            <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgerse…
## $ bill_length_mm    <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, …
## $ bill_depth_mm     <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, …
## $ flipper_length_mm <int> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186…
## $ body_mass_g       <int> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, …
## $ sex               <fct> male, female, female, NA, female, male, female, male…
## $ year              <int> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…

From this summary, we can see there are 344 rows (each row is one penguin), 8 columns, and we see the first few values in each column. Another way to explore the raw data in RStudio is to run

View(penguins)

in your console (not in R Markdown!).

Other functions can also be used to calculate the number of rows, the number of columns, and both the number of rows and number of columns:

nrow(penguins) # number of rows

## [1] 344

ncol(penguins) # number of columns

## [1] 8

dim(penguins) # dimensions of the data

## [1] 344   8

Missing data

Sometimes, our data contains missing values. These are often recorded as NA in R (for “not available”). You can see when we glimpsed the penguins data above that there were several missing values (NAs), and there are probably more in the rest of the data that wasn’t displayed.

To remove missing rows which contain missing values, we can use the drop_na function. The following code creates a new dataset (called penguins_new) without those missing values:

penguins_new <- penguins %>%
  drop_na()

How many rows did we remove (i.e., how many rows had missing values)? We can compare the dimensions of penguins_new to penguins:

dim(penguins_new)

## [1] 333   8

Since penguins_new has 333 rows, we have removed 11 rows due to missing values.

penguins_new <- penguins %>%
  drop_na(flipper_length_mm)

dim(penguins_new)

## [1] 342   8

A note on pipes (%>% and |>)

Throughout this codebook (and throughout my own code too!), I use the pipe operator %>%. The pipe just means “take THIS, then do THAT”. So,

penguins %>%
  drop_na(flipper_length_mm)

means “take penguins, then drop_na (remove missing values)”. Pipes can be chained together into a longer sequence of steps, too:

penguins %>%
  drop_na(flipper_length_mm) %>%
  dim()

## [1] 342   8

I like pipes because I think they often make code cleaner and more readable, but you don’t have to use them. You can collapse the pipe into a nested series of functions (evaluated from the inside out):

dim(drop_na(penguins, flipper_length_mm))

## [1] 342   8

Or you can save the intermediate steps:

penguins_new <- drop_na(penguins, flipper_length_mm)
dim(penguins_new)

## [1] 342   8

The pipe operator %>% comes from the magrittr package (the name is a nod to artist Rene Magritte and his painting The Treachery of Images). Newer versions of R also include a built-in pipe operator |> which functions similarly:

penguins |>
  drop_na(flipper_length_mm) |>
  dim()

## [1] 342   8

For simplicity, and because |> is more recent, in this course I will default to the older pipe %>%.

Types of variables

Here we will deal mainly with categorical and quantitative variables.

• Categorical variables are variables which take on one of several fixed values. These values generally do not have a numeric interpretation.
  • Examples: gender, favorite food, brand of laptop
  • Binary categorical variables have exactly 2 possible values
• Quantitative variables are variables which take on a numeric value, and which have a numeric interpretation.
  • Examples: number of pets, height, weight, age
  • Discrete quantitative variables only take on discrete values (e.g., number of pets)
  • Continuous quantitative variables can take on an entire range of values (e.g., height is continuous if we allow heights like 60.323 inches)

An overview of ggplot for making plots

The ggplot2 package, which is part of tidyverse, is a very valuable tool for visualizing data. The ggplot2 packages provides the ggplot function, which is my default for making plots. For example, here is code to create a scatterplot:

penguins %>%
  ggplot(aes(x = bill_depth_mm, 
             y = bill_length_mm, 
             color = species)) +
  geom_point() +
  labs(x = 'Bill depth (mm)', 
       y = 'Bill length (mm)',
       color = 'Species',
       title = "Penguin bill length vs. bill depth") +
  theme_bw()

## Warning: Removed 2 rows containing missing values (geom_point).

Notice that ggplot is warning us that we had some missing values in the data. If we remove those missing values, the warning will go away.

penguins %>%
  ggplot(aes(x = bill_depth_mm, 
             y = bill_length_mm, 
             color = species))

penguins %>%
  ggplot(aes(x = bill_depth_mm, 
             y = bill_length_mm, 
             color = species)) +
  geom_point()

## Warning: Removed 2 rows containing missing values (geom_point).

penguins %>%
  ggplot(aes(x = bill_depth_mm, 
             y = bill_length_mm, 
             color = species)) +
  geom_point() +
  labs(x = 'Bill depth (mm)', 
       y = 'Bill length (mm)',
       color = 'Species',
       title = "Penguin bill length vs. bill depth") +
  theme_bw()

## Warning: Removed 2 rows containing missing values (geom_point).

Univariate exploratory data analysis

Here we discuss how to summarize, visualize, and describe the distributions of categorical and quantitative variables.

penguins %>%
  count(species)

## # A tibble: 3 × 2
##   species       n
##   <fct>     <int>
## 1 Adelie      152
## 2 Chinstrap    68
## 3 Gentoo      124

penguins %>%
  ggplot(aes(x = species)) +
  geom_bar() +
  labs(x = "Species") +
  theme_bw()

penguins %>%
  summarize(mean_mass = mean(body_mass_g, na.rm=TRUE),
            median_mass = median(body_mass_g, na.rm=TRUE),
            sd_mass = sd(body_mass_g, na.rm=TRUE),
            iqr_mass = IQR(body_mass_g, na.rm=TRUE))

## # A tibble: 1 × 4
##   mean_mass median_mass sd_mass iqr_mass
##       <dbl>       <dbl>   <dbl>    <dbl>
## 1     4202.        4050    802.     1200

penguins %>%
  ggplot(aes(x = body_mass_g)) +
  geom_histogram(bins = 20) +
  labs(x = "Body mass (g)") +
  theme_bw()

penguins %>%
  ggplot(aes(y = body_mass_g)) +
  geom_boxplot() +
  labs(y = "Body mass (g)") +
  theme_bw()

Bivariate exploratory data analysis

What if we want to look at the relationship between two variables?

penguins %>%
  count(species, island)

## # A tibble: 5 × 3
##   species   island        n
##   <fct>     <fct>     <int>
## 1 Adelie    Biscoe       44
## 2 Adelie    Dream        56
## 3 Adelie    Torgersen    52
## 4 Chinstrap Dream        68
## 5 Gentoo    Biscoe      124

penguins %>%
  count(species, island) %>%
  spread(island, n)

## # A tibble: 3 × 4
##   species   Biscoe Dream Torgersen
##   <fct>      <int> <int>     <int>
## 1 Adelie        44    56        52
## 2 Chinstrap     NA    68        NA
## 3 Gentoo       124    NA        NA

penguins %>%
  ggplot(aes(x = species, fill = island)) +
  geom_bar() +
  labs(x = "Species",
       fill = "Island") +
  theme_bw()

penguins %>%
  ggplot(aes(x = flipper_length_mm, 
             y = body_mass_g)) +
  geom_point() +
  labs(x = "Flipper length (mm)",
       y = "Body mass (g)") +
  theme_bw()

penguins %>%
  summarize(r = cor(flipper_length_mm, 
                    body_mass_g, 
                    use="complete.obs"))

## # A tibble: 1 × 1
##       r
##   <dbl>
## 1 0.871

penguins %>%
  ggplot(aes(x = species, y = body_mass_g)) +
  geom_boxplot() +
  labs(x = "Species", y = "Body mass (g)") +
  theme_bw()

penguins %>%
  ggplot(aes(x = body_mass_g)) +
  geom_histogram(bins = 20) +
  facet_wrap(~species) +
  labs(x = "Body mass (g)") +
  theme_bw()

penguins %>%
  group_by(species) %>%
  summarize(mean_mass = mean(body_mass_g, na.rm=TRUE),
            median_mass = median(body_mass_g, na.rm=TRUE))

## # A tibble: 3 × 3
##   species   mean_mass median_mass
##   <fct>         <dbl>       <dbl>
## 1 Adelie        3701.        3700
## 2 Chinstrap     3733.        3700
## 3 Gentoo        5076.        5000

More than two variables

With more than two variables, you can get a lot of combinations. Here are just a couple examples. Using additional aesthetics and faceting is helpful for visualization. Using grouping is helpful for summary statistics.

penguins %>%
  ggplot(aes(x = bill_depth_mm, 
             y = body_mass_g, 
             color = species)) +
  geom_point() +
  labs(x = "Bill depth (mm)",
       y = "Body mass (g)",
       color = "Species") +
  theme_bw()

penguins %>%
  group_by(species) %>%
  summarize(r = cor(bill_depth_mm, 
                    body_mass_g, 
                    use="complete.obs"))

## # A tibble: 3 × 2
##   species       r
##   <fct>     <dbl>
## 1 Adelie    0.576
## 2 Chinstrap 0.604
## 3 Gentoo    0.719

penguins %>%
  ggplot(aes(x = island, 
             y = body_mass_g)) +
  geom_boxplot() +
  facet_wrap(~species) +
  labs(x = "Island", 
       y = "Body mass (g)") +
  theme_bw()

penguins %>%
  ggplot(aes(x = body_mass_g)) +
  geom_histogram(bins=15) +
  facet_grid(island~species) +
  labs(x = "Body mass (g)") +
  theme_bw()

penguins %>%
  group_by(species, island) %>%
  summarize(mean_mass = mean(body_mass_g, na.rm=TRUE),
            median_mass = median(body_mass_g, na.rm=TRUE))

## # A tibble: 5 × 4
## # Groups:   species [3]
##   species   island    mean_mass median_mass
##   <fct>     <fct>         <dbl>       <dbl>
## 1 Adelie    Biscoe        3710.        3750
## 2 Adelie    Dream         3688.        3575
## 3 Adelie    Torgersen     3706.        3700
## 4 Chinstrap Dream         3733.        3700
## 5 Gentoo    Biscoe        5076.        5000

Two components of a parametric model

In the penguins data, sex is a categorical variable which is recorded as either male or female. Suppose we want to predict penguin sex based on physical characteristics like flipper length and body mass. Let \(Y_i = 0\) if sex = female, and \(Y_i = 1\) if sex = 1 (I am coding male as 1 here because by default, R will order the levels of sex alphabetically, so female and then male). Then we might propose the following population logistic regression model:

\[Y_i \sim Bernoulli(\pi_i)\] \[\log \left( \dfrac{\pi_i}{1 - \pi_i} \right) = \beta_0 + \beta_1 FlipperLength_i + \beta_2 BodyMass_i\] Here \(\pi_i\) denotes the probability that \(Y_i = 1\), and the log odds are a linear function of flipper length and body mass.

The first line of this model is called the random component: it specifies the distribution of the random variable \(Y_i\). The second line is called the systematic component: it specifies how \(\pi_i\) is related to the explanatory variables.

Fitting logistic regression models

To fit the above logistic regression model in R, we use the glm function:

m1 <- glm(sex ~ flipper_length_mm + body_mass_g, 
          data = penguins, family = binomial)

summary(m1)

## 
## Call:
## glm(formula = sex ~ flipper_length_mm + body_mass_g, family = binomial, 
##     data = penguins)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.9901  -0.9598   0.2442   0.8898   2.1068  
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)        7.0868039  2.6621921   2.662  0.00777 ** 
## flipper_length_mm -0.0932568  0.0199664  -4.671 3.00e-06 ***
## body_mass_g        0.0027820  0.0003923   7.092 1.32e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 461.61  on 332  degrees of freedom
## Residual deviance: 371.98  on 330  degrees of freedom
##   (11 observations deleted due to missingness)
## AIC: 377.98
## 
## Number of Fisher Scoring iterations: 4

The glm function stands for generalized linear model. We specify the response variable on the left hand side of the ~, and the explanatory variables on the right hand side. To fit logistic regression, we specify family = binomial.

The summary function summarizes our fitted model. We can see that \(\widehat{\beta}_0 = 7.087\), \(\widehat{\beta}_1 = -0.093\), and \(\widehat{\beta}_2 = 0.003\). Our fitted logistic regression model is therefore

\[Y_i \sim Bernoulli(\pi_i)\] \[\log \left( \dfrac{\widehat{\pi}_i}{1 - \widehat{\pi}_i} \right) = 7.087 -0.093 FlipperLength_i + 0.003 BodyMass_i\]

Note that the equation of the fitted model includes hats on the \(\pi_i\).

Interpreting regression coefficients

The fitted coefficients above can be interpreted on the log odds scale, or on the odds scale.

Log odds:

• The estimated log odds that a penguin is male when flipper length and body mass are both 0 is 7.087. (Since all penguins have flippers and no penguins weigh 0g, this interpretation doesn’t mean much)
• A one mm increase in flipper length is associated with an estimated decrease in the log odds that a penguin is male by 0.093, holding body mass constant
• A one g increase in body mass is associated with an estimated increase in the log odds that a penguin is male by 0.003, holding flipper length constant

Odds:

• The estimated log odds that a penguin is male when flipper length and body mass are both 0 is \(\exp\{7.087\} = 1196\). (Since all penguins have flippers and no penguins weigh 0g, this interpretation doesn’t mean much)
• A one mm increase in flipper length is associated with an estimated decrease in the log odds that a penguin is male by a factor of \(\exp\{-0.093\} = 0.911\), holding body mass constant
• A one g increase in body mass is associated with an estimated increase in the log odds that a penguin is male by a factor of \(\exp\{0.003\} = 1.003\), holding flipper length constant

Empirical logit plots

The following R function can be used to create empirical logit plots:

logodds_plot <- function(data, num_bins, bin_method,
                         x_name, y_name, grouping = NULL, 
                         reg_formula = y ~ x){
  
  if(is.null(grouping)){
    dat <- data.frame(x = data %>% pull(x_name), 
                      y = data %>% pull(y_name),
                      group = 1)
  } else {
    dat <- data.frame(x = data %>% pull(x_name), 
                      y = data %>% pull(y_name),
                      group = as.factor(data %>% pull(grouping)))
  }
  
  if(bin_method == "equal_size"){
    logodds_table <- dat %>%
      drop_na() %>%
      arrange(group, x) %>%
      group_by(group) %>%
      mutate(obs = y,
             bin = rep(1:num_bins,
                       each=ceiling(n()/num_bins))[1:n()]) %>%
      group_by(bin, group) %>%
      summarize(mean_x = mean(x),
                prop = mean(c(obs, 0.5)),
                num_obs = n()) %>%
      ungroup() %>%
      mutate(logodds = log(prop/(1 - prop)))
  } else {
    logodds_table <- dat %>%
      drop_na() %>%
      group_by(group) %>%
      mutate(obs = y,
             bin = cut(x, 
                       breaks = num_bins,
                       labels = F)) %>%
      group_by(bin, group) %>%
      summarize(mean_x = mean(x),
                prop = mean(c(obs, 0.5)),
                num_obs = n()) %>%
      ungroup() %>%
      mutate(logodds = log(prop/(1 - prop)))
  }
  
  if(is.null(grouping)){
    logodds_table %>%
      ggplot(aes(x = mean_x,
                 y = logodds)) +
      geom_point(size=2) +
      geom_smooth(se=F, method="lm", formula = reg_formula) +
      theme_bw() +
      labs(x = x_name,
           y = "Empirical log odds") +
      theme(text = element_text(size=15))
  } else {
    logodds_table %>%
      ggplot(aes(x = mean_x,
                 y = logodds,
                 color = group,
                 shape = group)) +
      geom_point(size=2) +
      geom_smooth(se=F, method="lm", formula = reg_formula) +
      theme_bw() +
      labs(x = x_name,
           y = "Empirical log odds",
           color = grouping,
           shape = grouping) +
      theme(text = element_text(size=15))
  }
  
}

Arguments:

• data: the dataset of interest (e.g., titanic)
• num_bins: the number of bins to use
  • The number of bins should be chosen based on the size of the data. For example, with bin_method = "equal_size", we probably want at least 15 observations per bin, so num_bins < (number of observations)/15
• bin_method: whether to choose bins with the same number of observations ("equal_size") or the same width ("equal_width")
• x_name: the name of the column containing the explanatory variable (e.g., "Fare"). The quotation marks are needed
• y_name: the name of the column containing the response (e.g., "Survived"). The quotation marks are needed
• grouping: the name of a categorical variable in the data; fit a separate line for each level of the categorical variable
• reg_formula: This is the shape of the relationship you want to plot. If you want a line, this is y ~ x (the default). Some other examples:
  • y ~ log(x) : a log transformation of x
  • y ~ sqrt(x) : a square root transformation of x
  • y ~ poly(x, 2) : a second-order polynomial
  • y ~ poly(x, 3) : a third-order polynomial

library(tidyverse)
logodds_plot(titanic, 30, "equal_size", "Fare", "Survived",
             reg_formula = y ~ x)

logodds_plot(titanic, 30, "equal_size", "Fare", "Survived",
             reg_formula = y ~ poly(x, 2))

logodds_plot(titanic, 30, "equal_size", "Fare", "Survived",
             reg_formula = y ~ log(x))

Quantile residual plots

Quantile residual plots can be created using the qresid() function of the statmod package. For example, suppose we fit a logistic regression on the titanic data with Survival as the response and Fare as the explanatory variable:

m1 <- glm(Survived ~ Fare, data = titanic, family = binomial)

Let’s assess whether the shape assumption is reasonable for Fare with a quantile residual plot:

library(statmod)
titanic %>%
  mutate(residuals = qresid(m1)) %>%
  ggplot(aes(x = Fare, y = residuals)) +
  geom_point() +
  geom_smooth() +
  theme_bw()

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

The results aren’t amazing…what if we try transforming Fare? Let’s try a square root:

m2 <- glm(Survived ~ sqrt(Fare), data = titanic, family = binomial)
titanic %>%
  mutate(residuals = qresid(m2)) %>%
  ggplot(aes(x = sqrt(Fare), y = residuals)) +
  geom_point() +
  geom_smooth() +
  theme_bw()

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Looks a bit better!

vectors

A vector is a simple way of storing multiple items. For example, here is a vector containing the numbers 2, 1, 4:

v <- c(2, 1, 4)
v

## [1] 2 1 4

The c() in R means “combine” or “concatenate” the elements of the vector together. The items inside the parentheses are the elements of the vector.

length(v)

## [1] 3

v[1]

## [1] 2

v[2]

## [1] 1

v[3]

## [1] 4

w <- c('a', 'b', 'c', 'd')
w

## [1] "a" "b" "c" "d"

c(v, w)

## [1] "2" "1" "4" "a" "b" "c" "d"

seq(from=0, to=1, by=0.1)

##  [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

seq(from=1, to=10, by=1)

##  [1]  1  2  3  4  5  6  7  8  9 10

1:10

##  [1]  1  2  3  4  5  6  7  8  9 10

rep(0, 11)

##  [1] 0 0 0 0 0 0 0 0 0 0 0

for loops

Suppose we want to repeat a process many times (e.g., simulate many sets of data). A for loop allows us to do this efficiently. Here is a simple for loop which prints out the number 214 five times:

for(i in 1:5){
  print(214)
}

## [1] 214
## [1] 214
## [1] 214
## [1] 214
## [1] 214

for(...) indicates that we are making a for loop. The code inside the curly braces { ... } is the code which will get repeated. The index i here tells us how many times we run this code. Remember that 1:5 is R shorthand for 1,2,3,4,5. So i in 1:5 means “run the code inside the loop for i=1, i=2, i=3, i=4, and i=5”.

The really neat thing about a for loop is that what we do inside the loop can depend on the index i! For example, suppose we want to print out the numbers 1 to 5, instead of 214:

for(i in 1:5){
  print(i)
}

## [1] 1
## [1] 2
## [1] 3
## [1] 4
## [1] 5

pis <- seq(from=0, to=1, by=0.1) # values of pi to consider
likelihoods <- c() # empty vector to store the likelihoods we calculate
for(i in 1:length(pis)){
  # likelihood = pi*(1 - pi)^4, for each value of pi
  likelihoods[i] <- pis[i]*(1 - pis[i])^4
}

likelihoods

##  [1] 0.00000 0.06561 0.08192 0.07203 0.05184 0.03125 0.01536 0.00567 0.00128
## [10] 0.00009 0.00000

pis <- seq(from=0, to=1, by=0.1)
likelihoods <- pis*(1 - pis)^4

likelihoods

##  [1] 0.00000 0.06561 0.08192 0.07203 0.05184 0.03125 0.01536 0.00567 0.00128
## [10] 0.00009 0.00000

Sum and product notation

At its core, statistics relies on math, and math often uses notation to make equations and expressions simpler. Some notation that we will often encounter in STA 214 is sum (\(\sum\)) and product (\(\prod\)) notation.

Suppose we have \(n\) observations, \(Y_1,...,Y_n\). We want to take the sum of these observations: \[Y_1 + Y_2 + Y_3 + \cdots + Y_n\] Great! But this requires us to write out each of the terms, or use \(\cdots\) to express that there are terms we haven’t written out. Regardless, it is a little clunky. We can make this neater by using \(\sum\) as a shorthand for “sum” (\(\Sigma\) is the capital sigma in Greek, so think “S for Sum”). Using this notation, \[\sum \limits_{i=1}^n Y_i = Y_1 + Y_2 + \cdots + Y_n\] This shorthand means “sum up all the \(Y_i\), starting with \(i = 1\) and going to \(i = n\)”. In other words, sum up \(Y_1,...,Y_n\).

What about products? The product of \(Y_1,...,Y_n\) is \[Y_1 Y_2 \cdots Y_n\] which we can write more cleanly as \(\prod \limits_{i=1}^n Y_i\). Here \(\prod \limits_{i=1}^n\) just means “multiply these \(n\) things together”. (\(\Pi\) is the capital pi in Greek, so think “P for Product”)

Logs and properties of logs

Recall that logarithms (logs for short) are what we use to undo exponents. That is, \(\log(x) = y\) means that \(x = e^{y}\). (By default, \(\log\) in math and statistics is the natural log. Equivalent definitions exist for other bases: \(\log_{10}(x) = y\) means that \(x = 10^{y}\)).

There are some useful properties of logs that we use in STA 214, particularly for maximum likelihood estimation:

• Logs turn products into sums: \[\log(a \cdot b) = \log(a) + \log(b)\] More generally, \[\log \left( \prod \limits_{i=1}^n a_i \right) = \sum \limits_{i=1}^n \log(a_i)\] Likewise, \[\log(a/b) = \log(a) - \log(b)\]

• Logs turn exponents into multiples: \[\log(x^a) = a \log(x)\]

• Logs have nice derivatives: \[ \dfrac{d}{dx} \log(x) = \dfrac{1}{x}\]

Calculus review: differentiation

Recall the following rules for derivative:

• \(\dfrac{d}{dx} c f(x) = c \dfrac{d}{dx} f(x)\) for constant \(c\) More generally
• \(\dfrac{d}{dx} (f(x) + c) = \dfrac{d}{dx} f(x)\) for constant \(c\)
• The derivative of a sum is the sum of derivatives: \[\dfrac{d}{dx} (f(x) + g(x)) = \dfrac{d}{dx} f(x) + \dfrac{d}{dx} g(x)\] More generally, this means that derivatives can move inside \(\sum\): \[\dfrac{d}{dx} \sum \limits_{i=1}^n f_i(x) = \sum \limits_{i=1}^n \dfrac{d}{dx} f_i(x)\]
• Chain rule: \[\dfrac{d}{dx} f(g(x)) = f'(g(x)) g'(x)\]